Applied Mechatronics and Mechanics: System Integration and Design 177188889X, 9781771888899

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APPLIED MECHATRONICS

AND MECHANICS

System Integration and Design

APPLIED MECHATRONICS

AND MECHANICS

System Integration and Design

Edited by Satya Bir Singh, PhD

Prabhat Ranjan, PhD

A. K. Haghi, PhD

First edition published 2021 Apple Academic Press Inc. 1265 Goldenrod Circle, NE, Palm Bay, FL 32905 USA 4164 Lakeshore Road, Burlington, ON, L7L 1A4 Canada

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© 2021 Apple Academic Press, Inc. Apple Academic Press exclusively co-publishes with CRC Press, an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the authors, editors, and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors, editors, and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library and Archives Canada Cataloguing in Publication Title: Applied mechatronics and mechanics : system integration and design / edited by Satya Bir Singh, PhD, Prabhat Ranjan, PhD, A.K. Haghi, PhD. Names: Singh, Satya Bir, editor. | Ranjan, Prabhat, (Mechatronics professor), editor. | Haghi, A. K., editor. Description: Includes bibliographical references and index. Identifiers: Canadiana (print) 20200305255 | Canadiana (ebook) 20200305433 | ISBN 9781771888899 (hardcover) | ISBN 9781003019060 (PDF) Subjects: LCSH: Mechatronics. | LCSH: Mechanics, Applied. Classification: LCC TJ163.12 .A67 2021 | DDC 621—dc23 Library of Congress Cataloging‑in‑Publication Data

CIP data on file with US Library of Congress

ISBN: 978-1-77188-889-9 (hbk) ISBN: 978-1-00301-906-0 (ebk)

About the Editors

Satya Bir Singh, PhD

Professor, Department of Mathematics, Punjabi University, Patiala, India Satya Bir Singh, PhD, is a Professor of Mathematics at Punjabi University Patiala in India. Prior to this, he worked as an Assistant Professor in Math­ ematics at the Thapar Institute of Engineering and Technology, Patiala, India. He has published about 125 research papers in journals of national and international repute and has given invited talks at various conferences and workshops. He has also organized several national and international conferences. He has been a coordinator and principal investigator of several schemes funded by the Department of Science and Technology, Government of India, New Delhi; the University Grants Commission, Government of India, New Delhi; and the All India Council for Technical Education, Government of India, New Delhi. He has 21 years of teaching and research experience. His areas of interest include the mechanics of composite materials, optimization techniques, and numerical analysis. He is a life member of various learned bodies.

Prabhat Ranjan, PhD

Assistant Professor in the Department of Mechatronics Engineering at Manipal University Jaipur, India Prabhat Ranjan, PhD, is an Assistant Professor in the Department of Mechatronics Engineering at Manipal University Jaipur, India. He is the author of the book Basic Electronics and editor of the book Computational Chemistry Methodology in Structural Biology and Materials Sciences. Dr. Ranjan has published more than 10 research papers in peer-reviewed journals of high repute and dozens of book chapters in high-end edited research books. He has received prestigious the President Award–Manipal University Jaipur, India, in 2015 for the development of the university; a Material Design Scholarship from Imperial College of London, UK,

vi

About the Editors

in 2014; a DAAD Fellowship in 2014; and CFCAM–France Award in 2015. Dr. Ranjan has received several grants and has also participated in national and international conferences and summer schools. He holds a Bachelor of Engineering in Electronics and Communication and a Master of Technology in Instrumentation Control System Engineering from the Manipal Academy of Higher Education, Manipal, India; as well as PhD in engineering from Manipal University Jaipur, India.

A. K. Haghi, PhD

Professor Emeritus of Engineering Sciences, Former Editor-in-Chief, International Journal of Chemoinformatics and Chemical Engineering and Polymers Research Journal; Member, Canadian Research and Development Center of Sciences and Culture A. K. Haghi, PhD, is the author and editor of 165 books, as well as 1000 published papers in various journals and conference proceedings. Dr. Haghi has received several grants, consulted for a number of major corporations, and is a frequent speaker to national and international audi­ ences. Since 1983, he served as professor at several universities. He is former Editor-in-Chief of the International Journal of Chemoinformatics and Chemical Engineering and Polymers Research Journal and is on the editorial boards of many international journals. He is also a member of the Canadian Research and Development Center of Sciences and Cultures (CRDCSC), Montreal, Quebec, Canada. He holds a BSc in urban and environmental engineering from the University of North Carolina (USA), an MSc in mechanical engineering from North Carolina A&T State University (USA), a DEA in applied mechanics, acoustics and materials from the Université de Technologie de Compiègne (France), and a PhD in engineering sciences from Université de Franche-Comté (France).

Contents

Contents Contributors ......................................................................................................... ix

Abbreviations ....................................................................................................... xi

Preface ............................................................................................................... xiii

1.

Dynamic Balancing of Planar Mechanisms Using Nondominated Sorting Jaya Algorithm .............................................................................. 1 Prem Singh, Ramanpreet Singh, and Himanshu Chaudhary

2.

Vibration Control of a Car Suspension System Using a Magnetorheological Damper with a Fuzzy‑Logic Controller ............ 17 G. Pohit

3.

A Review of the Application of Mechatronics in Manufacturing Processes .................................................................................................... 41 T. Muthuramalingam

4.

Application of a Nature‑Inspired Algorithm in Odor Source Localization ............................................................................................... 51 Kumar Gaurav, Ramanpreet Singh, Ajay Kumar, and Ram Dayal

5.

Tire Pressure Monitoring Systems: A Case Study in Automotive Mechatronics ........................................................................ 67 Narayan Kumar and B. Ravindra

6.

A Retrospective Assessment of Elastic‑Plastic and Creep Deformation Behavior in Structural Components Like Discs, Cylinders, and Shells ................................................................................ 81 Shivdev Shahi, Satya Bir Singh, and A. K. Haghi

7.

Investigation of Elastic‑Plastic Transitional Stresses in Zirconia‑Based Ceramic Dental Implants under Uniaxial Compression .............................................................................................. 97 Shivdev Shahi, Satya Bir Singh, and A. K. Haghi

Contents

viii

8.

Recent Developments in the Theory of Nonlocality in Elastic and Thermoelastic Mediums...................................................................111 Sukhveer Singh, Parveen Lata, and Satya Bir Singh

9.

Investigation of Creep Performance of an Isotropic Composite Disc Under Thermal Gradients ............................................................. 123 Vandana Gupta and Satya Bir Singh

10. Application of a Fractional‑Order PID Controller to an Underactuated System............................................................................ 137 Tung-Yung Huang and Shih-Ying Hung

11. Evaluating the Effect of an Amputee’s Physical Parameters of Pressure on a Lower‑Limb Prosthetic Socket Using a Fuzzy‑Logic‑Based Model ...................................................................... 167 Vimal Kumar Pathak, Chitresh Nayak, and Deepak Rajendra Unune

12. Experimental Investigation and Optimization of Process Parameters in Oblique Machining Process for Hard‑to‑Cut Materials Using Coated Inserts ............................................................. 191 Purnank Bhatt, Mihir Solanki, Anand Joshi, and Vijay Chaudhari

13. Mechanical Design of a Slider‑Crank Mechanism for a Knee Orthotic Device Using the Jaya Algorithm ................................. 209 Ramanpreet Singh and Vimal Kumar Pathak

Index.................................................................................................................. 223

Contributors

Purnank Bhatt

Assistant Professor, Mechanical Engineering Department, G. H. Patel College of Engineering & Technology, Vallabh Vidyanagar, Gujarat, India

Vijay Chaudhari

Professor, Mechanical Engineering Department, C. S. Patel Institute of Technology, Charusat University, Changa, Gujarat, India

Himanshu Chaudhary

Mechanical Engineering Department, Malaviya National Institute of Technology, Jaipur, Rajasthan, India

Ram Dayal

Department of Mechanical Engineering, Malaviya National Institute of Technology, Jaipur, Rajasthan, India

Kumar Gaurav

Department of Mechatronics Engineering, Manipal University Jaipur, Dehmi Kalan, Jaipur, Rajasthan, India, E-mails: [email protected]; [email protected]

Vandana Gupta

Department of Mathematics, Dashmesh Khalsa College, Zirakpur, Mohali, Punjab, India, E-mail: [email protected]

A. K. Haghi

Professor Emeritus, Canadian Research and Development Center of Sciences and Cultures, Montreal, Canada

Tung‑Yung Huang

Department of Mechanical Engineering, Southern Taiwan University of Science and Technology, Tainan, Taiwan, E-mail: [email protected]

Shih‑Ying Hung

Department of Mechanical Engineering, Southern Taiwan University of Science and Technology, Tainan, Taiwan

Anand Joshi

Professor, Mechatronics Engineering Department, G. H. Patel College of Engineering & Technology, Vallabh Vidyanagar, Gujarat, India

Ajay Kumar

Department of Mechatronics Engineering, Manipal University Jaipur, Dehmi Kalan, Jaipur, Rajasthan, India

Narayan Kumar

Department of Mechanical Engineering, IIT Jodhpur, Karwar, Jodhpur – 342037, Rajasthan, India

x

Contributors

Parveen Lata

Associate Professor, Department of Basic and Applied Science, Punjabi University Patiala, Punjab, India

T. Muthuramalingam

Department of Mechatronics Engineering, SRM Institute of Science and Technology, Kattankulathur – 603203, Tamil Nadu, India, E-mail: [email protected]

Chitresh Nayak

Professor, Oriental Institute of Science and Technology, Bhopal, Madhya Pradesh, India

Vimal Kumar Pathak

Assistant Professor, Department of Mechanical Engineering, Manipal University Jaipur, Dehmi Kalan, Jaipur – 303007, Rajasthan, India

G. Pohit

Department of Mechanical Engineering, Jadavpur University, Kolkata – 700032, West Bengal, India, E-mail: [email protected]

Prabhat Ranjan

Assistant Professor in the Department of Mechatronics Engineering at Manipal University Jaipur, India, E-mail: [email protected]

B. Ravindra

Department of Mechanical Engineering, IIT Jodhpur, Karwar, Jodhpur – 342037, Rajasthan, India, E-mail: [email protected]

Shivdev Shahi

Research Scholar, Department of Mathematics, Punjabi University, India, E-mail: [email protected]

Prem Singh

Mechanical Engineering Department, Swami Keshvanand Institute of Technology, Jaipur, Rajasthan, India, E-mail: [email protected]

Ramanpreet Singh

Department of Mechanical Engineering, Manipal University Jaipur, Dehmi Kalan, Jaipur, Rajasthan, India, E-mail: [email protected]

Satya Bir Singh

Professor, Department of Mathematics, Punjabi University, Patiala, Punjab, India, E-mail: [email protected]

Sukhveer Singh

Assistant Professor, Punjabi University APS Neighborhood Campus, Dehla Seehan, Sangrur, Punjab, India, E-mail: [email protected]

Mihir Solanki

Assistant Professor, Mechanical Engineering Department, G. H. Patel College of Engineering & Technology, VallabhVidyanagar, Gujarat, India

Deepak Rajendra Unune

Assistant Professor, Department of Mechanical Engineering, The LNM Institute of Technology, Jaipur, Rajasthan, India

Abbreviations

ADC ANOVA BFO BMVSS BWSAS COG COTS CRONE DAQ DF ECU EDM FFS FGMs FG-MWCNTs FSR GA H HAP HIL HSS HT HTLPSO KAFO L LEER M MCMC MGA MOOP M-PSO MR NHTSA

analog to digital converter analysis of variance bacterial foraging optimization Bhagwan Mahaveer Viklang Sahayata Samiti Bouc-Wen-based semi-active suspension system centroid of gravity commercially off the shelf Commande Robuste d’Ordre Non-Entier data acquisition degree of freedom engine control unit electrical discharge machining flexible force sensor functionally graded materials functionally graded multi-walled carbon nanotubes force-sensing resistors genetic algorithm high hydroxyapatite hardware-in-loop high-speed steel height hybrid-teaching learning particle swarm optimization knee-ankle-foot orthoses low lower extremity exoskeleton robot medium Markov chain Monte Carlo modified genetic algorithm multi-objective optimization problem modified particle swarm optimization magnetorheological National Highway Traffic Safety Administration

xii

NSJAYA OSL PaS PGO PID PSO PTB RF RMSE SDE SiCp SIL SL SUBAR TPMS TPMS VA VH VL WT

Abbreviations

nondominated sorting Jaya algorithm odor source localization passive suspension system powered gait orthosis proportional-integral-derivative particle swarm optimization patella tendon bearing radio frequency root mean square error source declaration error silicon carbide particles software-in-loop stump length Sogang University biomedical assistive robot tire pressure maintenance system tire pressure monitoring sensor virtual agents very high very low weight

Preface

Mechatronics is a core subject for engineers, combining elements of mechanical and electronic engineering into the development of computercontrolled mechanical devices. A mechatronics system integrates various technologies involving sensors, measurement systems, drives, actuation systems, microprocessor systems, and software engineering. A universally accepted definition of the term “mechatronics” is the integration of several disciplines such as mechanics, electronics, electrical, computer, control, and software engineering using microelec­ tronics to control mechanical devices. Applied mechatronics and mechanics is the integration of mechanical engineering, electronic engineering, control, and computer engineering. Synergistic collaboration among these fields of science involves a high potential for accomplishments and achievements now accessible to a wide variety of engineers. This research-oriented book presents a clear and comprehensive introduction to the area and helps the readers to comprehend and design mechatronic systems by providing a frame of understanding to develop a truly interdisciplinary and integrated approach to engineering. The design of mechatronics products is a challenge for design engi­ neers. Therefore, the mechatronic process is a cross-disciplinary design process that can be properly applied if, and only if, the specialists from all pertinent disciplines work together from a very early stage in the design process. Applied Mechatronics and Mechanics is the most comprehensive research-oriented book available for both mechanical and electrical engineering students and will enable them to engage fully with all stages of mechatronic system design. It offers broader and more integrated coverage than other research-oriented books in the field with case studies. This volume is a source of the latest research and technical notes in mechatronics. This book is useful for students, researchers, and all readers interested in this topic.

xiv

Preface

This volume focuses on application considerations and relevant practical issues that arise in the selection and design of mechatronics components and systems as well. It provides practicing mechanical/mechatronics engineers and designers, researchers, graduate, and postgraduate students with knowl­ edge of mechanics focused directly on advanced applications. Chapter 1 presents the balancing procedure of the mechanism using a multi-objective Jaya algorithm. The mechanisms can be balanced by optimizing the inertial properties of each link. The inertial properties of each rigid link are represented by the dynamic equivalent system of point masses. Thus, the multi-objective optimization problem (MOOP) with the minimization of shaking forces and shaking moments is formu­ lated by considering the point mass parameters as the design variables. The formulated optimization problem is solved by a posteriori approachbased algorithm as a multi-objective Jaya algorithm (MOJAYA) under suitable design constraints. This algorithm uses the concepts of crowding distance and a nondominated sorting approach to find a Pareto set of optimal solutions. The efficiency of the proposed approach is investigated through a four-bar planar standard mechanism taken from literature. It is established that the multi-objective Jaya algorithm is computationally more efficient than NSGA-II. The designer can choose any solution from the set and balance any real planar mechanisms. In Chapter 2, different types of suspension systems are discussed. The advantage of a semi-active suspension system is highlighted. It is shown that MR damper is an important part of automotive suspension systems and, hence, plays a major role in controlling the dynamic nature of the vehicle. The effectiveness of the MR damper was studied. Initially, a quarter car model with a passive suspension system (PaS) was designed. Another quarter car model was developed with an MR damper as a member of the semi-active suspension system. The Bouc-Wen hyster­ esis model predicted the behavior of the MR damper. The final quarter car model was developed with an MR damper based on the Bouc-Wen model and controlled by a fuzzy-logic controller. These three models were analyzed by considering the step road profile. Altering the values of the scalar gains to produce better results optimized the fuzzy-logic controller. In Chapter 3, the importance of enhancing the performances of conventional manufacturing processes is being increased. It is possible

Preface

xv

only where integrating the mechatronics approaches in manufacturing processes. The adaptation of control algorithms and modification of existing electronics circuits in the manufacturing processes can consid­ erably enhance the performances of the manufacturing technology. The efficiency of the unconventional machining processes can be effectively increased by the mechatronics engineering concepts. In the present study, an endeavor has been made to analyze the various literature related to the adaptation of mechatronics knowledge in manufacturing technology. The responses of the processes with and without mechatronics approaches have been compared and the merits of those systems over the existing systems have been analyzed. The influences of the various control strat­ egies have also been analyzed and compared. It has been observed that the still better research works can be made to increase the efficacy the manufacturing technology. Chapter 4 applies the nature-inspired optimization-based approach for odor source localization (OSL) problem in an indoor environment with mobile agents. An optimization problem is formulated to identify the region of maximum odor concentration. The optimization problem is solved using a new hybrid-teaching learning particle swarm optimization (HTLPSO) algorithm. To demonstrate the effectiveness of the adopted approach, experiments are conducted in a simulated environment gener­ ated through the Gaussian plume model. A various set of mobile agents in a range of {3–15} are used to find and detect the source of odor with high accuracy and concentration, respectively. Furthermore, it is found that less number of mobile agents is required for fast convergence to detect the odor source. Tire pressure monitoring systems as a case study in automotive mechatronics presented in Chapter 5. Chapter 6 is a retrospective assess­ ment of elastic-plastic and creep deformation behavior in structural components made of discs, cylinders, and spherical/cylindrical shells. Unlike elastic solids in which the state of strain depends only on the final state of stress, the deformation that occurs in a plastic solid is determined by the complete history of loading. The plasticity problem is, therefore, essentially incremental in nature, the final distortion of the solid being obtained as the sum total of the incremental distortions following the strain path. The deformation character has been considered to be linear in classical theory which is not accurate. The non-linear character was studied by B. R. Seth considering the transition surface function and

xvi

Preface

generalized strain measure to determine elastic-plastic and creep state. A number of elastic-plastic and creep problems pertaining to various structural components, made of materials exhibiting different kinds of isotropy and anisotropy have been solved using this transition theory and are reviewed in this chapter. It is sufficient to say that the transition functions, which define the non-linearity of elastic-plastic and creep transition, are more accurate when compared to the classical theory. In Chapter 7, elastic-plastic transition stresses in Zirconia-based crowns of ceramic dental implants have been calculated analytically. The crown of the implant is modeled in the form of a shell which exhibits transversely isotropic macro-structural symmetry. The transition theory of B. R. Seth has been used to model the elastic-plastic state of stresses. The shell so modeled is subjected to external pressure to analyze the state of axial compression. The results for Zirconia-based implants are compared with a titanium-based dental implant. The elastic stiffness constants for these are taken from the available literatures which have been obtained using ultrasonic resonance spectroscopy, a non-destructive technique of obtaining the stiffness constants. The radial and circumfer­ ential stresses are obtained for radius ratios, which can handle any type of dataset for thicknesses of crowns. Recent developments in the theory of nonlocality in elastic and ther­ moelastic mediums; presented in Chapter 8. In Chapter 9, a mathematical model is developed to investigate the creep response in rotating composite disc with hyperbolic thickness for isotropic material. The disc is supposed to be made of the composite containing silicon carbide particles (SiCp) embedded in a matrix of pure aluminum. The model is used to investigate the effect thermal gradients on the stresses and strain rates of the isotropic disc by Sherby’s creep law and von Mises criterion of yielding. It is concluded that thermal gradients introduce a significant change in the strain rates although its effect on the resulting stress distribution is relatively small in a composite rotating disc. The aim of Chapter 10 is to find an adequate rule of tuning frac­ tional-order PID controllers for dynamic systems. PID controllers of integer order have been successfully employed in versatile applications, especially with the aid of available mature tuning methods such as the Ziegler-Nichols method. In recent years, fractional-order PID controllers tuned with the Ziegler-Nichols type method were proposed to deal with

Preface

xvii

fractional-order systems. It is intriguing for the flexibility of the adop­ tion of fractional order which is free from the limit of integer order. This article proposes a Ziegler-Nichols type tuning method for the fractional order PID controller applied in an underactuated system, and studies the effect of fraction order. The simulation results demonstrate that the proposed method works effectively. Chapter 11 presents a methodology for evaluating the role of ampu­ tee’s physical parameters viz. height, weight, and stump length (SL) on the pressure generated at the prosthetic limb/socket interface using the regression technique and fuzzy-logic-based model. Intra-socket pressures can cause the tissue trauma or discomfort to amputees wearing the pros­ thetic devices. The intra-socket pressure values at lateral tibia, gastroc­ nemius, patella tendon bearing (PTB), kick point, medial tibia, medial gastrocnemius, popliteal depression, and lateral gastrocnemius on the transtibial residual limb have been collected for three different conditions viz. walking, full load and half load for ten patients. An experimental setup is developed for force investigation of the lower-limb socket using the Flexi-Force sensor. The experimental trials are performed and further experimental data are used to establish the Mamdani fuzzy-logic model to predict the effective pressure at considered specific regions as the response parameters. Mathematical models for pressure at three loading conditions have been developed using regression analysis using which the effective pressure at considered specific regions can be correlated with physical parameters. The models suggested that the weight and followed by stump length of amputees are a strong predictor of pressure at the socket. The confirmation experiment results reveal that the fuzzy model shows good agreement of 98.53% with the experimentally meas­ ured value that proves that the established fuzzy model consequently can be used for predicting the effective pressure at the socket interface for different points. The developed methodology will assist the ampu­ tee-specific socket design ensuring comfortable socket fitting. Experimental investigation and optimization of process parameters in oblique machining process for hard to cut materials coated inserts presented in Chapter 12. Chapter 13 presents the mechanical design of a slider-crank mecha­ nism for a knee joint orthotic device. The device is portable for stroke patients and helps in gait training at home, clinics, and hospitals. The knee joint mechanism is optimized by considering gait biomechanics for

xviii

Preface

making it potable. An optimization problem is formulated by considering the geometrical parameters of slider-crank linkage with an objective to reduce the required peak force by an actuator while walking on a leveled ground. Based on the optimization problem, constraints are posed to ensure the range of motion as in normal human gait. The optimization problem is solved using the Jaya algorithm. It is observed that there is a significant reduction in the peak force required by the actuator.

CHAPTER 1

Dynamic Balancing of Planar Mechanisms Using Nondominated Sorting Jaya Algorithm

PREM SINGH,1 RAMANPREET SINGH,2 and HIMANSHU CHAUDHARY3 Mechanical Engineering Department, Swami Keshvanand Institute of Technology, Jaipur, Rajasthan, India, E-mail: [email protected]

1

Department of Mechanical Engineering, Manipal University Jaipur, Rajasthan, India

2

Mechanical Engineering Department, Malaviya National Institute of Technology, Jaipur, Rajasthan, India

3

ABSTRACT This chapter presents the balancing procedure of the mechanism using a nondominated sorting Jaya algorithm (NSJAYA). Planar mechanisms can be balanced by optimizing each moving link’s inertial properties. These properties are derived by the point-mass system. Thus, the multi-objective optimization problem (MOOP) is posed to minimize the unbalance force and moment in which the point mass parameters are treated as the design variables. The formulated optimization problem is solved by a posteriori approach-based algorithm as a NSJAYA under suitable design constraints. This algorithm uses the concept of crowding distance and a nondominated sorting approach to find a Pareto set of optimal solutions. The proposed method is tested through a four-bar planar standard mechanism taken from literature. It is established that the NSJAYA is computationally more efficient than NSGA-II. The designer can choose any solution from the set and balance any real planar mechanisms.

2

Applied Mechatronics and Mechanics: System Integration and Design

1.1 INTRODUCTION Typically, the unbalanced mechanism develops the forces and moments on the bearings which are also known as shaking forces and moments. They increase the vibration, driving torque, fatigues, etc., in the mechanism. Inertia, the center of mass, and mass of moving link define the input torque, the forces, and moments [1]. Recently, the balancing of these forces and moments is a challenging task. Therefore, many techniques have been applied to reduce the forces and moments using various principles such as counterweights [2] and the redistribution of the masses [3] are used to minimize the forces. But, this force balancing procedure generally increases moments and input torque of the mechanism [4]. Moreover, disk or inertia counterweights [5, 6], moment balancing idler loops [7], and a replicate mechanism [8] are used to minimize the moments. These approaches of balancing enhance the mass and complicacy of the mechanism [4]. In contrast, the optimization procedure has also been used by researchers to balance the mechanisms. Typically, two objectives, namely, shaking force and shaking moment are chosen. Thus, the optimization problem with multi-objectives is posed to minimize these objectives [9]. The formulated problem can be solved using two approaches as a priori approach and a posteriori approach [10]. The first approach obtains a single objective problem from the multi-objective problem using an appropriate weight for each objective function. This approach gives a unique optimal design in each simulation run. Therefore, multiple optimal solutions are obtained with a different combination of weights. Moreover, the optimum results obtained by this approach depend upon the weights assigned to each objective function. Therefore, the designer must know the importance of each objective function while assigning the weights to the objective functions that can be difficult for an uncertain scenario. In addition, a posteriori approach eliminates the drawbacks of a priori approach. In this approach, the weights are not assigned to the objectives before the start of the algorithm. It provides Pareto optimal solutions in the single run of the algorithm. The user can choose appropriate solutions based on the importance of objective functions from the Pareto optimal set of solutions [11]. This makes the posteriori approach computationally more efficient in comparison to the priori approach. Therefore, it is applied for the balancing problems of the planar mechanism. The conventional optimization techniques like gradient search methods based on a priori approach can be applied to balance the shaking force

Dynamic Balancing of Planar Mechanisms

3

and moment [12, 13]. However, they need an initial start point to find the optimal solution [14]. Thus, these algorithms give the local solution near to start point. Moreover, nature-inspired optimization techniques like the genetic algorithm (GA) [15, 16] and particle swarm optimization (PSO) [17], and a hybrid of two optimization algorithms [18] can be applied to balance the mechanisms. But, these optimization techniques require specific parameters for their convergence. No relevant research has been published in which a posteriori approach based algorithm is applied to balance the mechanisms. In this study, a posteriori approach based algorithm as a nondominated sorting Jaya algorithm (NSJAYA) is applied to balance the planar mechanism. The shaking forces and shaking moments are defined using the concept of the dynamically equivalent point mass system [19]. This system represents the inertial properties of each moving link. In order to balance the mecha­ nism, the optimization problem with multi objectives as shaking force and shaking moment are stated by considering the parameters of point masses as the design variables. The efficiency of the proposed method is validated through a standard four-bar planar mechanism taken from the literature. It is found that the NSJAYA algorithm is more efficient in computation than NSGA-II used in this study. The Pareto optimal set for the balancing problem is calculated and outlined in this chapter. The designer can balance any real planar mechanisms using any solution from the Pareto optimal set of solutions. This chapter is structured as follows: Section 1.2 describes the dynamic analysis of the planar mechanism; Section 1.3 describes the optimization problem formulation using the concept of a point mass system; Section 1.4 presents the optimization algorithm. Results and discussion are outlined in Section 1.5; and finally, Section 1.6 outlines the conclusions. 1.2 DYNAMIC ANALYSIS OF THE PLANAR MECHANISM This section presents the determination of the shaking forces and shaking moments developed in the four-bar planar mechanisms as shown in Figure 1.1. {o1 xy} and {oi xi yi} are the global and local coordinate systems, respectively. The reaction forces at joints are determined using NewtonEuler equations [20]. Then, these forces determine the shaking force and shaking moment. The shaking force is the summation of transmitted forces to fixed link in vector form [21] while the vector sum of driving torques

4

Applied Mechatronics and Mechanics: System Integration and Design

and moments of the reaction forces about the fixed points is known as shaking moment [22].

FIGURE 1.1

Representation of reaction forces and moments.

Thus, shaking forces and shaking moments are transferred to the fixed link ‘#0,’ given as:

f sh

F01 F03

(1.1)

M sh

T01 r0 F03

(1.2)

where F01 and F03 are 2-vector reaction forces of link #1 and link #3 at link #0, respectively, while T01 is driving torque about joint o1. r4 is the vector from o1 to o4. 1.3 OPTIMIZATION PROBLEM FORMULATION This section formulates the optimization problem with multi objectives as the shaking force and shaking moment using a point-mass system

Dynamic Balancing of Planar Mechanisms

5

approach. The mechanism shown in Figure 1.1 has three moving links. Each moving link defined as in Figure 1.2(a) is systematically converted into the system of three-point masses as shown in Figure 1.2(b). The point-mass parameters are treated as the design variables. These param­ eters describe the shaking force and moment which are reported in Eqs. (1.1)–(1.2). Generally, three parameters need to specify each point mass while each link needs nine parameters to represent it. The vector form of i-th link’s design variables is represented as:

xi

mi1 i1ai1mi 2

a mi 3

i2 i2

a

T

i3 i3

(1.3)

where mij is the j-th point mass for the i-th link, aij and θij are the corre­ sponding length and angular position of mij. Thus, a total of 27 point mass parameters defines the mechanism, given as:

x = [x1T x2T x3T ]T

(1.4)

The multi-objective optimization problem (MOOP) is finally stated under the appropriate constraints by taking the R.M.S values of shaking force, Fsh,rms and the shaking moment, Msh,rms as: Minimize

f1 x

Fsh ,rms

(1.5)

Minimize

f2 x

M sh ,rms

(1.6)

Subjected to:

g1i ( x) = mi, min - " j mij